What is Wrong with an Algorithm?

Nothing! Yes I said nothing! Let me elaborate.

In conversation today I was asked, "What is wrong with an algorithm?"  I get this question a lot. Now this question was asked because like us all we were taught to do an algorithm so we always do what is familiar for us. The question also came about because they were afraid of teaching their child the wrong way.

This inspired me to write this blog post. There is nothing wrong with learning an algorithm; there are some risks in only learning this but there is nothing wrong.

An algorithm was first invented in order to make math easier to do. We have to remember that calculators were not invented at this time and may people were using abacuses and basically counting by ones or a base 16 system. With the use of an algorithm counting became procedural and easier to do.  However, those using it understood mathematical foundations of why the algorithm worked.

This is the problem with introducing the algorithm to early. Let's just reflect on our own understanding of basic addition. We honestly did not learn how to add by using an algorithm first and to be fair our first insistence of remembering an algorithm (at least in my guess) was probably around grade three maybe four. This means that for four-five years, depending if you were in the pre-school or JK error you had a lot of experimentation and exploring in counting, saying numbers, ordering numbers and subitizing.  The problem is that we often forget all of the steps it took us in learning an algorithm.

For the past ten years I have been looking closely at how students learn mathematics. One of the foundation research pieces have been Fosnot and Dolk's landscape of Learning (2000). (full file)

I know that the picture is small but you can see all of the learning that a student has to go through in order to get to an understanding of algorithms.

Now here is my problem with starting with algorithms to early:

1) It doesn't teach proper number sense:

Now you may question me on this but in my honest opinion it doesn't teach proper number sense. The problem is that when a person does an algorithm it forces you to only look at numbers from 1-9. There is no true understanding of our base ten system. Students do not really understand why or how 1 group of ten can be ten things yet 1 ten. This is often seen when you ask a student tell you the value of a two digit number (let's say 24). When you point to the ten's column they will say the value is two. They will even go as far as counting 24 objects then pulling two of them to signify the value. This can also be seen when students use base ten blocks and they count the rods as two instead of twenty.

2) Basic procedural errors:

Now these errors may be dismissed as, they just have to learn the proper procedure. However, these are troubling errors because students not only are procedurally doing the operations wrong they are also struggling with reason-ability of answers or what I like to call no number sense.  

What I propose instead:

In my math class we work hard on developing a number line.  Learning this strategy does a lot of things: 

1) It allows all students to grasp a strategy whether they are counters by one or able to skip count proficiently. 

2) It teachers all of the rudimentary learning of number sense. Students learn one to one tagging, cardinality, subitizing, magnitude and many more options all in one strategy.

3) Eventually they develop more proficient mental strategies because they can conceptually understand what is happening to the numbers.

4) There is no need to regroup and or borrow with a number line, just the use of number sense.  I find this to be the biggest problem with algorithms. For example, let's take 56-49=7. Now for many of us we may not need to use an algorithm but ask your child or a younger student they will start by borrowing from the 5 (not fifty) to make 16 because you cannot take 6 from 9 (which we also no is not true-- ask anyone in debt or who has a mortgage) and then they will realize the answer is five. However, if students are taught proper number sense, they realize that they can just count up to the answer.

A number line also develops efficient mental math strategies that are far faster then whipping out a piece of paper and borrowing or regrouping.  Take a look at some of my grade twos thinking (all done 

Our number talk today. Continuing to build adding and subtracting by ten. #raylearns #engagemath #peelmathchat pic.twitter.com/gIp8HnOnXv
— Jonathan So (@MrSoclassroom) October 31, 2014

Our number talk today. Trying to get Ss to think w/ 10s. http://t.co/Sd45gMG2Do #enagemath #peelmatchat #raylearns pic.twitter.com/pHI7hNd7in
— Jonathan So (@MrSoclassroom) October 29, 2014

By allowing my students to explore, question and develop their number sense they are better mathematicians and yes we have discussed an algorithm, we have developed an understanding of how to use it but my students make less mistakes with a number line then with an algorithm. And to be honest they much prefer the number line then the algorithm, most of the time they do this because that is what they think they need to do.

One of the biggest problems in our math programs today is that we often jump to far into the abstract without thinking about the concrete work that people need to develop in order to understand these concepts.

What do you think? When and where should algorithms be taught? How do you develop number sense? Love to hear your thoughts.

For further reading I recommend reading:

1) Fosnot, C. T., & Dolk, M. L. A. M. (2001). Young mathematicians at work. Portsmouth, NH: Heinemann.

2) Anghileri, J., Beishuizen, M., & Van Putten, K. (2002). From informal strategies to structured procedures: mind the gap!. Educational Studies in Mathematics,49(2), 149-170. 

3) Kami and Dominck: The Harmful effects of Algorithms in 4-9

Growth versus Fix Mindset: An #Engagemath PD reflection

Yesterday was a Professional Development Day for our Board (Peel). All of the schools (minus the balance calendar schools) where all busy engaged in mathematics and talking about effective mathematics instruction. However, the over arching theme was more about a growth mindset versus a fixed. Though I am in a balance calendar I was able to participate in Burnt Elm's PD session and thought I would jot down some thoughts about what we learned.  The session was run by Jason Wigmore (@Jaywigmore), Jackie Brown, Jane Hedi-Knapp and Judy Hyndman. I was ask to help with the planning and fielding the questions about math. Quite a new experience.

Now the title of the blog is Growth versus Fixed mindsets and our board has taken that approach for mathematics. Many of us grew up with the thought that math is a skill that only a few can handle, you either got it or didn't but this is a fixed mindset. Yes certain topics can be hard for students but that doesn't mean we cannot achieve understanding.

Take a look at this video:

if it doesn't work: https://www.youtube.com/watch?v=btDHUHZ6fAw 

What are your thoughts of the video?

In Peel or at least at the two schools I have done PD at this has been connected to the various evolution or reforms in mathematical teaching. For many (if not all) of us mathematics was taught a lot differently then it is now. This is very scary for some and often challenges our ideals and schemas. When these believes are challenged is when we often get resistance. At both Ray Lawson (had ours first week of Sept) and Burnt Elm, we challenged the participates to think about their own schemas and believes and to adapt a growth mindset to their learning.

Our session started with a review of Stein, Engle, and Smith's Orchestrating Productive Mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show and Tell. We actually talked about their book but this article sums it up quite nicely. If you read the article would love to know your thoughts? What spoke to you about this article? What resonated as the most important point? 

I then was asked to tell a personal testimony of my mathematical journey.  I know that I have been preaching reform for some time now but I have not always felt this way.  Before I became a teacher I volunteered at a school who was just starting using reform.  For me this was a shock to the system. Math for me was never hard, in fact at this point in time I was taking my University Calculus class. I felt that if algorithms where invented then they should be used. I also strongly felt that steps and procedures could easily teach students how to do math. I was lucky enough to be pushed by an amazing principal who allowed me to question the process, give me research and see it in action. Its interesting to look back at this and see that really I was discovering these ideals through a constructivist approach: I had to explore, have a mentor and then experiment with my findings.  As time went on I noticed how students were achieving amazing results in mathematics and sustaining them but more importantly their attitudes towards math changed.  I saw all students engaged in lessons, learning math and being mathematicians.  The more I taught the more I saw these observations consistently happening.  Not only this but while teaching at Brookmede I was able to be a part of a school which consistently implemented Reform mathematics.  During this time not once did I have to review last years concepts with the students, I started each year with that years concepts. Yes I had students who were developmentally not ready for the grade but they all could tackle the problems, worked together collaboratively and loved math. To me that was enough.  I share this story because if I had a fixed mind-set and believed that the way I was taught is the best way then I wouldn't have seen the benefit of teaching through problem solving. Now that being said I am always revisiting my teaching practise and I think we always have to be. Growth is growth! Change must happen.

The session then turned into a great problem solving lesson.  One in which the teachers got to feel that disequilibrium that comes with learning mathematics.

“@MrSoclassroom: Some great math talk happening here. Grt job burnt elm and @JayWigmore pic.twitter.com/uTa0Zksb20” #EngageMath
— Jason Wigmore (@JayWigmore) October 10, 2014 

After debriefing the problem we had a great discussion from a teachers perspective. The questions we discussed where:

1) Why were the strategies chosen in the debrief?

This brought out a cool discussion about levels or accessibility to the problem and talk.  A lot of the times when strategies are chosen its because we want to honour student voice but in reality we do have to think about the math.  The first strategy should be one that can have the most students engaged with the lesson. One that allows the most talk and will lead to the next progression of the lesson. It was during this time too that we addressed the thought that our goal should never be to get kids to jump from a basic understanding right away to the abstract, this is what causes the gaps in learning.  It is far richer to move them up to the next stage in development. To do this as a teacher we have to think through the possibilities and understand what our students may or may not accomplish.

2) What is the role of a teacher in this format?

3) What is the impact on the student?

After this session, the principal open the discussion up for general questions.  It was great to hear all of the questions being asked and it reminded me of the journey that I went through.  Let me highlight a few of these questions:

1) Are you telling me Algorithms are bad?
No I am suggesting that there is a lot of understanding that goes into learning algorithms that many of our elementary students do not have.  We need to think about the middle piece that goes into the learning of math.  We often forget as adults all of the processes that we had to go through to learn a concept.

2) What about facts?
I know that I have answered this question before and it always gets me that people associate Reform with not teaching facts. In fact as we have discussed in #engagemath there is a balance and yes FACTS ARE IMPORTANT. One needs facts to solve math but how we go about learning facts is a different matter.  Jo Boaler suggests that fluency does not mean speed. In fact I would suggest that speed is a fixed mind set. A mind set that can often discourage children from learning math.  I teach facts through games which does three things. One it teaches fact recall. Two it teaches communication, understanding and strategy and three it build relationships. Here is a link to my math games.  

3) What is the most important aspect of setting up a classroom like this?
Classroom setup is vitally to building problem solvers.  Students need to feel comfortable, held accountable and know that mistakes are learning opportunities.  I always find that the best way to do this is wait.  Waiting time is so under utilized as a teacher. We want to fill the void but if you just wait one more minute longer you will be surprise at what students say. Here is some things that I have collected on math talk.  

Overall,  it was an amazing PD session at both Burnt Elm and Ray Lawson (though Ray Lawsons did not happen on Friday). Peel teachers continue to be amazing both in their thinking and sharing.  When I do talk about math I am reminded the learning that I had to go through and I know that we are all in various spots but keep that growth mindset not only for your self but your students too. That being said I would love to here your thoughts on anything discussed here:

1) Reform Mathematics

2) Growth versus Fixed Mindsets

3) Teaching through problem solving

4) Five practises

Also if you have any questions or comments please contact me or leave them below.

Fraction kit and playing games

Fractions have always been a passion of mine. Started researching the concepts in my math part 1 AQ class and have been fascinated ever since.  I even ended up completing my Masters' of Education thesis in the subject.  Through my studies I came across fractions, Marilyn Burns' fraction kit and games.  I still haven't found something anywhere close that helps students understand fraction concepts like this kit.  

For those not familiar with it, let me tell you about it.  The kit in itself is very simple, it is five strips of paper. Each piece is to be cut to a corresponding fraction (halves, quarters, eighths, sixteenths, and a whole). 

Now you may ask yourselves how is this the best thing ever it's just a bunch of paper. It's the best thing ever because of the talk that it generates. Since finding this in my research I have done some modifications that really bring out the talk. 

First and foremost, I have them create the kits. It does you no good to create them for your students.  By them creating the strips, the students explore how fractions are division, fair sharing, why fractions are a part of a whole and many more fractional concepts.

Second, I created a context to go with the problem. As many of you know who read this blog, I truly believe in contexts. A good context makes kids think beyond arithmetic and focus on mathematical big ideas.  For this problem I tell my students a story of how I need to clean up my mom's back yard, she has a huge yard and in payment my mom buys me a large party sub. Now many students now don't know what a party sub is because they don't sell them anymore, so you may have to show them a picture: 

The students are so impressed and they can't believe that I would eat this much. Now I tell them that just before I was about to eat lunch one of my friends popped over. Now what?  This continues all the way to eights, the door bell ringing every time we figure out portion we need to cut.  For sixteenths I tell them this is what we are going to do as I really don't have sixteen friends; however by now we have really constructed a good understanding of the pattern that is happening.  Now why this context. I like this context because it is a linear model like the strips. Having the sub also means students have to think about measurement and division because technically you cannot fold a sub, as all the pieces fall out. The other part is students often will try cutting the their strips horizontally instead of vertically. Now this also brings up interesting discussions about equivalency versus congruency but this context stops that because if students cut a sub horizontally they don't really get all of the sub.

Third I don't have the students label their fractions.  When I have done this with my fours it was mainly because I didn't want them to associate a particular fraction with the strips whole. Basically, 1/2 strip is 1/2 of the kits whole not 1/2 somewhere else.  A big misconception with students thinking is that a what they learn is he only representation of a particular fraction. When you label the students don't understand that the size of the whole matters.  That 1/4 can be bigger than 1/2 depending on the size.  However, now that I am in primary I see a whole new benifit, it makes students understand what a fraction is. Why is 1/4, 1/4? While my students where playing cover-up, one of Mariyln burns fraction kit games, they asked me which fraction is 1/4? I turned it around and asked them. They then just picked a random strip up. I the. Asked them why that one? This discussion continued as students explored that the amount of pieces that we break our sub into is our denominator and the amount we use is our numerator.   If I had them label the fractions they never would have explored this concept and I would never have realized that they struggled with it.

The final change is the questions that I ask around this particular problem.  It's not just to make the stud ets create the kit but to think about the big ideas around fractions. Have a listen to my grade two class discussion on fractions:

Day 1 of our Fraction Talk

Day 2 of the Talk:

It is quite interesting the talk that can come from building these kits and the big ideas that come from it. I have played this game in junior and primary and personally I would do this for middle school as well.  In junior I start to add fifths, tenths, thirds, sixths, ninths, and twelves.  By adding these other fractions you also start to see other misconceptions of students halving strategies but for primary halving is still okay.  I hope you really try the kits and see the benefits of it in your classroom.

You can find all of my fraction research and resources on my site: Bit.ly/Soresources.  Feel free to use anything you want.

Why do we Need to Argue over Math? --> A Call for a Balanced Approach

About a month ago a colleague of mine Kyle Pearce wrote a post "Does memorizing multiplication facts hurt more than help." It was a very interesting read and I happen to agree with Kyle's point of view.  As many of my frequent readers of this blog know, I prescribe to the constructivist approach to learning mathematics.  I believe that students through discovery and proper guidance will be able to understand a wide load of big ideas and theories.  Not only do I believe this but I have witnessed this first hand with my students in every grade that I have taught.

However, this is not so for many people.  In fact it was a discussion on Kyle's blog post (feel free to read the thread) that has me thinking more and more about this topic. And not only thinking about it but trying to fix and insight thoughtful discussion around the ways in which we are teaching math.

Maybe a little background first.  Math has been a hot topic for the past year, if not for the last century.  For many countries, provinces, and states, math curriculum has undergone a significant change from what we grew up with as children.  Some. like myself, believe that these changes are for the better, some have not.  I was recently at the OAME and listening to Brent Davis a professor at University of Calgary.  In his lecture he shared that the reason math curriculum was introduced was so we could have a work force to crunch numbers, nothing more and nothing less.  As we have evolved beyond that (not saying fact crunching is not important) our skills have also changed and I think this is what we need to remember; we have evolved.

For this reason I and many others are proposing a  more balanced approach to mathematics.  Lets stop this war and needless debates and get to teaching good mathematical practises.  One in which our students will push their thinking and really think about the numbers.

If it was up to me this is what I would include:

1) Math should be linked to Big Mathematical Ideas:

I think this is the first step to thinking about our students as mathematicians.  Catherine Fosnot (2002) has some very interesting work around making our students mathematicians.  One of the most interesting facts is there was a study done with so many mathematicians and they were asked to solve a problem.  Not one of those mathematicians solved it the same way.  I found this interesting because that is what I see math.  Math is about the mathematics and there is not one way of doing things.  We have to teach our students the understanding, the flexibility and the patience to be mathematicians.

2) Math is about real numbers:

Students need authentic experiences to learn.  Let's think about ourselves and how we learn.  Now some do learn through reading and replication but if you honestly think about how you learn a concept the best; it is through trail and error and than guidance from a mentor.  This is the same for our students.  They need real experiences so that they can play and discover the mathematical concepts.  In my personal experience both in tutoring high school students and teaching mathematics in the primary and junior divisions it's the contexts that allow students to really understand what they are doing.  It's the context that helps them build models of representation.  In my classroom, these are often done through social justice problems and real life contexts.

3) Students need time to explore:

This goes hand in hand with the above comment. As much as we need instruction, we also need exploration. Students need time to make mistakes, reflect, debate and discuss. These experiences allow students to make connections between concrete and abstract thinking. I was reminded at the OAME that every mistake makes a new synapse in the brain. We need these mistakes I order to solidify  our learning.

4) Students need Mentors:

My most recent research in understanding teachers questions has shown me the importance of teachers, not that I didn't believe that before.  With a shift towards discovery learning, we as educators have forgotten the importance of our role, or what even our role is.  I truly believe that we should not be the dispenser of knowledge but the mentor of that knowledge.  Though through exploration students will learn (many studies to show this) they may not have tools to reflect or pull together the big ideas.  I think this is where much of the back lash has come from discovery math, reform math or whatever you want to call it.  As students explore and discover there needs to be some sort of guidance.  This is where a teacher can shine and help students with the mathematics.  However, my research shows that for this to happen, a teacher needs to 1) have a good understanding of mathematics, 2) have a good understanding of how children learn mathematics and 3) plan. I know that all teachers plan but this planning involves thinking of big ideas, landscapes and possible questions.  It is these questions carefully placed that can allow students to figure out and make mathematical connections.

Take a look at this video of three of my students thinking about fractions.

They have never been taught fractions from me before this and in fact as a class we haven't even started the unit. However, that being said think about their learning and the role they play and the role that I play as a teacher.  Where do my questions come from? Why did I ask them at the time I did?

5) Time for debating, conjecturing, discussing and proving:

For me this is the time for a teacher to shine; however not in the traditional sense of standing in front of the class and lecture or tell students how it should be done. Just like students need time to explore they also need time to debate as a community. It is through this debate that students defend their thinking, conjecture, question and solidify their learning. Moreover, it goes beyond just showing and telling.  As a teacher the types of strategies that you show matter. How are you building the learning up, what questions are you modeling? How are you focusing the talk? How are you fostering talk? These are all questions that a teacher needs to be asking.

A great article to read is: https://drive.google.com/file/d/0B4245QONE7HaSExrSUlWTjBrNEU/edit?usp=sharing 

Also take a look at my most recent grade two conversation of multiplication.

6) Repeated Practise:

Yes I said repeated practice! Students need it, but it's not just doing procedural learning over and over again.  When I say repeated practise I mean a similar problem for students to continue their exploration.  Students, well most students, cannot solidify their learning through one experience.  In a typical unit my students will solve about 7-8 problems that could take three to four weeks to learn.  These problems build their learning and knowledge from day to day.

7) Skills:

Yes skills are important. They are needed but they are not needed like we use to think about it. For me it's how are we introducing facts. Do we make students think through facts? Are they taught in isolation or allong with concepts?  In my classroom, students practise facts at home, they play math games in the classroom (here is a file of my math games:  https://drive.google.com/folderview?id=0B4245QONE7HaaHl3M3ZKNWd3SUk&usp=drive_web). In addition before my problems start I often use string lessons which builds on mental math strategies and learning how to be flexible thinkers and playing with numbers. This to me is more important then struck memorization. It teaches students that numbers are not confined facts but that you can pull apart numbers and use known facts to solve other facts. Through this process students often learn all of their math facts, can recall them and use them in problems, which to me is way more important.

These are just a few of my thoughts on what I am calling a balanced math approach. I have a few more  to hash out around integrating and  implementing a center approach within my problem solving approach.

What are your thoughts?  Don't you think it is better to discuss and fix our problems rather than lay blame about which is better?  Shouldn't we think about our students first and their needs in their 21st century world?  Love to here your thoughts.